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Metamath Proof Explorer

Theorem cdlemk35

Description: Part of proof of Lemma K of Crawley p. 118. cdlemk29-3 with shorter hypotheses. (Contributed by NM, 18-Jul-2013)

Ref Expression
Hypotheses cdlemk4.b 
|- B = ( Base ` K )
cdlemk4.l 
|- .<_ = ( le ` K )
cdlemk4.j 
|- .\/ = ( join ` K )
cdlemk4.m 
|- ./\ = ( meet ` K )
cdlemk4.a 
|- A = ( Atoms ` K )
cdlemk4.h 
|- H = ( LHyp ` K )
cdlemk4.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk4.r 
|- R = ( ( trL ` K ) ` W )
cdlemk4.z 
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk4.y 
|- Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) )
cdlemk4.x 
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) )
Assertion cdlemk35 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> X e. T )

Proof

Step Hyp Ref Expression
1 cdlemk4.b 
 |-  B = ( Base ` K )
2 cdlemk4.l 
 |-  .<_ = ( le ` K )
3 cdlemk4.j 
 |-  .\/ = ( join ` K )
4 cdlemk4.m 
 |-  ./\ = ( meet ` K )
5 cdlemk4.a 
 |-  A = ( Atoms ` K )
6 cdlemk4.h 
 |-  H = ( LHyp ` K )
7 cdlemk4.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk4.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk4.z 
 |-  Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10 cdlemk4.y 
 |-  Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) )
11 cdlemk4.x 
 |-  X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) )
12 eqid 
 |-  ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
13 eqid 
 |-  ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) ) = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
14 eqid 
 |-  ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) ) G ) ) ) = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) ) G ) ) )
15 1 2 3 4 5 6 7 8 12 13 14 cdlemk34 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) ) G ) ) ) = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) ) ) ) )
16 9 oveq1i 
 |-  ( Z .\/ ( R ` ( G o. `' b ) ) ) = ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) )
17 16 oveq2i 
 |-  ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) )
18 10 17 eqtri 
 |-  Y = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) )
19 18 eqeq2i 
 |-  ( ( z ` P ) = Y <-> ( z ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) ) )
20 19 imbi2i 
 |-  ( ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) <-> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) ) ) )
21 20 ralbii 
 |-  ( A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) ) ) )
22 21 a1i 
 |-  ( z e. T -> ( A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) ) ) ) )
23 22 riotabiia 
 |-  ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) ) = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) ) ) )
24 11 23 eqtri 
 |-  X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) .\/ ( R ` ( G o. `' b ) ) ) ) ) )
25 15 24 eqtr4di 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) ) G ) ) ) = X )
26 1 2 3 4 5 6 7 8 12 13 14 cdlemk29-3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) ) G ) ) ) e. T )
27 25 26 eqeltrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> X e. T )